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          从工程角度学习卡尔曼滤波
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        <p>​        第一次接触卡尔曼滤波的时候，被它的五个核心公式给搞晕了，推导的过程也还蛮复杂的，可能当时推导出来了，过了一段时间又忘记了。这边就从工程人运用的角度去简单入门卡尔曼滤波，不去考虑其中复杂的数学公式。</p>
<p>​        卡尔曼有五个核心的公式，这些公式里面有一些符号和基本概念，我们需要提前了解，下面就简单介绍，其中引用了B站up主<a target="_blank" rel="noopener" href="https://space.bilibili.com/352976834%E8%A7%86%E9%A2%91%E8%AE%B2%E8%A7%A3%E4%B8%AD%E7%9A%84%E5%9B%BE%EF%BC%8C%E5%9C%A8%E8%BF%99%E9%87%8C%E5%AF%B9%E4%BB%96%E8%A1%A8%E7%A4%BA%E7%94%B1%E8%A1%B7%E7%9A%84%E6%84%9F%E8%B0%A2%E3%80%82">https://space.bilibili.com/352976834视频讲解中的图，在这里对他表示由衷的感谢。</a></p>
<h3 id="1-基本的滤波知识"><a href="#1-基本的滤波知识" class="headerlink" title="1.基本的滤波知识"></a>1.基本的滤波知识</h3><img src="https://xdl-blog-picture.oss-cn-shanghai.aliyuncs.com/img/微信截图_20210911142442.png" style="zoom:52%;">

<h3 id="2-卡尔曼直观图解"><a href="#2-卡尔曼直观图解" class="headerlink" title="2.卡尔曼直观图解"></a>2.卡尔曼直观图解</h3><p><img src="https://xdl-blog-picture.oss-cn-shanghai.aliyuncs.com/img/微信截图_20210911143323.png" style="zoom: 67%;"><img src="https://xdl-blog-picture.oss-cn-shanghai.aliyuncs.com/img/%E5%BE%AE%E4%BF%A1%E6%88%AA%E5%9B%BE_20210911145112.png"></p>
<p>​        图中纵坐标是概率密度，^代表估计值，不带^的是测量值。有横杠的代表先验估计值；没有横杠的是最优估计也叫做修正值，后验估计值。</p>
<p>​        最左边那个是上一时刻的最优估计值，就是卡尔曼滤波最终输出的值。中间那个是基于最左边那个估计出来的一个当前的估计值。最右边那个是当前时刻的观测值。那么当前时刻的最优估计值大概率是在下图中新画出来的地方，总结一下意思就是：当前的最优估计值是由先验估计和当前的观测值取公有的部分得到一个最优的估计值。</p>
<img src="https://xdl-blog-picture.oss-cn-shanghai.aliyuncs.com/img/微信截图_20210911145534.png" style="zoom:67%;">

<h3 id="3-卡尔曼公式简单理解"><a href="#3-卡尔曼公式简单理解" class="headerlink" title="3.卡尔曼公式简单理解"></a>3.卡尔曼公式简单理解</h3><p>​        实现过程：使用上一次的最优结果预测当前的值，(<strong>这个值叫做先验估计</strong>)</p>
<p>​                            同时使用观测值修正当前值，得到最优结果。</p>
<p><img src="https://xdl-blog-picture.oss-cn-shanghai.aliyuncs.com/img/%E5%BE%AE%E4%BF%A1%E6%88%AA%E5%9B%BE_20210911152409.png"></p>
<p>公式1：基于之前的最优估计来推出当前的先验估计。</p>
<p>公式2：是协方差公式，对应公式1先验估计的协方差。(先验估计值和真实值之间误差的协方差矩阵)</p>
<p>公式3：计算卡尔曼增益。</p>
<p>公式4：最优估计。加号左边那个是先验估计值，加号右边那个是观测值，Kt是一个权重，然后两者融合。</p>
<p>公式5：计算最优估计值和真实值之间误差的协方差矩阵。</p>
<p><strong>其中：Q和R是卡尔曼滤波器中主要需要调的东西。Q是过程噪声的方差，R是观测噪声的方差。</strong></p>
<h3 id="4-简单的matlab例子"><a href="#4-简单的matlab例子" class="headerlink" title="4. 简单的matlab例子"></a>4. 简单的matlab例子</h3><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><span class="line">close all; </span><br><span class="line">clear all;</span><br><span class="line"><span class="comment">%系统方程：X(k+1)=A*X(k)+w(k) </span></span><br><span class="line"><span class="comment">%观测方程：Z(k)=H*X(k)+v(k) </span></span><br><span class="line">A=<span class="number">1</span>; </span><br><span class="line">H=<span class="number">1</span>; </span><br><span class="line">X(<span class="number">1</span>)=<span class="number">25</span>; <span class="comment">%系统状态初始化 </span></span><br><span class="line"></span><br><span class="line">w=<span class="number">1</span>*<span class="built_in">randn</span>(<span class="number">100</span>,<span class="number">1</span>); <span class="comment">%系统噪声 </span></span><br><span class="line">v=<span class="number">0.1</span>*<span class="built_in">randn</span>(<span class="number">100</span>,<span class="number">1</span>); <span class="comment">%测量噪声 </span></span><br><span class="line">Q=cov(w); </span><br><span class="line">R=cov(v); </span><br><span class="line"></span><br><span class="line">Xe(<span class="number">1</span>)=<span class="number">25</span>; <span class="comment">%状态估计初值 </span></span><br><span class="line">Pe(<span class="number">1</span>)=<span class="number">0</span>; <span class="comment">%估计值与真实值之间的协方差矩阵初值 </span></span><br><span class="line"></span><br><span class="line"><span class="keyword">for</span> <span class="built_in">i</span>=<span class="number">2</span>:<span class="number">100</span> </span><br><span class="line">	X(<span class="built_in">i</span>)=A*X(<span class="built_in">i</span><span class="number">-1</span>)+w(<span class="built_in">i</span>); <span class="comment">% 【6】系统方程 </span></span><br><span class="line">	Z(<span class="built_in">i</span>)=H*X(<span class="built_in">i</span>)+v(<span class="built_in">i</span>);   <span class="comment">% 【7】观测方程 </span></span><br><span class="line">	</span><br><span class="line">	<span class="comment">%卡尔曼滤波五大核心方程 </span></span><br><span class="line">	Xp(<span class="built_in">i</span>)=A*Xe(<span class="built_in">i</span><span class="number">-1</span>);      <span class="comment">% 【1】根据滤波后最优估计值计算预测值 </span></span><br><span class="line">	Pp(<span class="built_in">i</span>)=A*Pe(<span class="built_in">i</span><span class="number">-1</span>)*A&#x27;+Q; <span class="comment">% 【2】计算预测值与真实值之间误差的协方差矩阵</span></span><br><span class="line">    K(<span class="built_in">i</span>)=Pp(<span class="built_in">i</span>)*H&#x27;/((H*Pp(<span class="built_in">i</span>)*H&#x27;+R));  <span class="comment">% 【3】计算卡尔曼增益 </span></span><br><span class="line">    Xe(<span class="built_in">i</span>)=Xp(<span class="built_in">i</span>)+K(<span class="built_in">i</span>)*(Z(<span class="built_in">i</span>)-H*Xp(<span class="built_in">i</span>)); <span class="comment">% 【4】计算滤波后估计值 </span></span><br><span class="line">    Pe(<span class="built_in">i</span>)=Pp(<span class="built_in">i</span>)-K(<span class="built_in">i</span>)*H*Pp(<span class="built_in">i</span>);        <span class="comment">% 【5】计算估计值与真实值之间误差的协方差矩阵 </span></span><br><span class="line"><span class="keyword">end</span> </span><br><span class="line"></span><br><span class="line">n=<span class="number">1</span>:<span class="number">100</span>; </span><br><span class="line">subplot(<span class="number">211</span>) </span><br><span class="line"><span class="built_in">plot</span>(n,Xp,<span class="string">&#x27;r&#x27;</span>,<span class="string">&#x27;linewidth&#x27;</span>,<span class="number">2</span>); <span class="comment">%绘制预测值 </span></span><br><span class="line"><span class="built_in">hold</span> on; </span><br><span class="line"><span class="built_in">plot</span>(n,Z,<span class="string">&#x27;b&#x27;</span>,<span class="string">&#x27;linewidth&#x27;</span>,<span class="number">2</span>); <span class="comment">%绘制测量值 </span></span><br><span class="line"><span class="built_in">hold</span> on; </span><br><span class="line"><span class="built_in">plot</span>(n,Xe,<span class="string">&#x27;g&#x27;</span>,<span class="string">&#x27;linewidth&#x27;</span>,<span class="number">2</span>); <span class="comment">%绘制估计值 </span></span><br><span class="line"><span class="built_in">hold</span> on; </span><br><span class="line"><span class="built_in">plot</span>(n,X-Xe,<span class="string">&#x27;k&#x27;</span>,<span class="string">&#x27;linewidth&#x27;</span>,<span class="number">2</span>); <span class="comment">%绘制误差 </span></span><br><span class="line"><span class="built_in">legend</span>(<span class="string">&#x27;预测值&#x27;</span>,<span class="string">&#x27;观测值&#x27;</span>,<span class="string">&#x27;滤波值&#x27;</span>,<span class="string">&#x27;误差&#x27;</span>) </span><br><span class="line">grid </span><br><span class="line">subplot(<span class="number">212</span>) </span><br><span class="line"><span class="built_in">plot</span>(n,K,<span class="string">&#x27;b&#x27;</span>,<span class="string">&#x27;linewidth&#x27;</span>,<span class="number">2</span>); <span class="comment">%绘制卡尔曼增益 </span></span><br><span class="line"><span class="built_in">legend</span>(<span class="string">&#x27;卡尔曼增益&#x27;</span>) </span><br><span class="line">grid</span><br></pre></td></tr></table></figure>
<img src="https://xdl-blog-picture.oss-cn-shanghai.aliyuncs.com/img/微信截图_20210911160357.png" style="zoom:67%;">

<p>​    仿真结果如上上图所示，不难发现，由于测量噪声相比于系统噪声小的多，所以估计值更倾向于观测值，此时的卡尔曼增益趋近于 1。</p>

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